SEOUL—The populist mayor of South Korea’s biggest metropolis said that he was is working to share Seoul’s brand of urban management with other global cities, but at the World Economic Forum in Davos this week he took a more cautious stance on sharing-focused technology companies such as Uber and Airbnb.

Park Won-soon, a 60-year-old former activist widely seen as an early leading candidate for South Korea’s presidency next year, has cast himself in the mold of New York City’s popular former Mayor Michael Bloomberg, using his...